black hole chemistry - dur€¦ · black hole chemistry durham student seminar wilson brenna...

Black Hole Chemistry

Durham Student Seminar Wilson Brenna January 19, 2014

Work with Robert Mann and Miok ParkUniversity of Waterloo

Table of Contents

1. Smarr Relations and Thermodynamics

2. Lifshitz-Symmetric Spacetimes

3. A Lifshitz Smarr Relation

5. Conclusion

4. Extension to Numerical Solutions

Smarr 1973PRL 30, 2

Electric potentialFirst Law of Thermodynamics:

Smarr Relation:

Smarr Relations and Thermodynamics

Let's look at the Reissner-Nordstrom black hole.

Does the first law hold?

How do we know we had the mass correct?

Komar Mass

Brown-York Quasilocal Mass

ADM Mass

Suggested refs: Poisson GR Text, 1402.1443


What happens when we add a cosmological constant?


What happens when we add a cosmological constant?

1. The metric requires another lengthscale ( ).

2. The previous Smarr relation may fail to hold.

First Law of Thermodynamics:


Smarr Relation:

Kastor, Ray, TraschenarXiv:0904.2765

Let's look at the Reissner-Nordstrom AdS black hole.

Maybe try adding the pressure term...

Why are we doing this, anyway?


and identify Gibbs free energy and Helmholtz free energy.

We can then find the equation of state:

Smarr gives the definition of pressure and TD volume:

(for Reissner-Nordström AdS Black Holes in 3+1 dimensions)

We want to compute thermodynamics and (eventually) universality classes.

Critical Exponents

order parameter


Kubiznak, MannarXiv:1205.0559

The 3+1 Dimensional Reissner-Nordström AdS Black Hole:


The Van der Waals Fluid:

Lifshitz-Symmetric Spacetimes

Anti de Sitter Spacetime

~ Conjectured dual to certain condensed matter systems

Lifshitz ~ Introduce an Anisotropy in Time

while


For example:


For example:

We don't know M or V. This is a problem!


Smarr obeys Eulerian Scaling:

Eulerian Scaling:


Smarr obeys Eulerian Scaling:

through the first law:


We obtain a nonlinear set of equations:


Is there an exact solution?

A Lifshitz Smarr Relation

Can we define a charge?


Can we define a charge?

Yes. It's a little more complicated, but proceeds in the same manner.


Can we define a mass via other methods?


Can we define a mass via other methods?

z=3, D=3

In some cases, yes!

e.g. Gim, Kim, Li; arXiv:1403.4704

(Brown-York mass with thermodynamically guessed-at counterterms)


Is the thermodynamic mass consistent?


Is the thermodynamic mass consistent?

For the unambiguous cases, yes.


Are our Smarr relation assumptions in agreement?


Other spacetimes for which our method agrees:

Reissner-Nordström AdS (z=1, D=4)

Non-rotating BTZ (z=1, D=3)

5D Lifshitz with Higher Curvature terms (z=2, D=5)

AdS-Taub-NUT (z=1, D=4)

Lifshitz with Einstein-Dilaton-Maxwell (z=D)


Dependent lenthscales yield ambiguous results.

4D Lifshitz with Maxwell Charge (z=4, D=4); Pang, arXiv:0901.2777

The problem here: horizon radius depends on the cosmologicallengthscale when q = 0. We have a number of ways to resolve this.

Create a fictitious mass in the metric function:

For p = (D+z-2), we obtain results that agree with a Wald formula:


If we have then these quantities are not independent.

Why does this happen?

We have a couple of options.



1. Manually separate the quantities in the metric.

2. Re-think how the action contributes new lengthscales.


1. Manually separate the quantities in the metric.


works in all cases we can check!



2. Re-think how the action contributes new lengthscales.


How come people use other Smarr relations?

A simplification occurs for planar black holes. In some cases:

while the two lengthscales are independent.

Via our method, the Smarr relation will simplify to


Via our method, the Smarr relation will simplify to

This is analogous to pretending pressure does not contribute(which is why this appeared for k = 0 black holes before 2009).In that case there is only one lengthscale, the horizon radius.

Extension to Numerical Solutions

This method works particularly well for numerical solutions.

If both series independently converge, do thermodynamic mass andvolume converge? Typically entropy is finite and known, so we require:

Conclusion

Next...

Anisotropy in Time ~ mass becomes a tricky concept

Lifshitz Smarr Relation ~ gives enoughinformation to obtain a TDmass and volume

Is a Maxwell field necessary for a finite-T critical point?

What is the universality class?

Attempt with numerical solutions!

Acknowledgments

Robert Mann

Miok Park

Sergey Solodukhin

University of Waterloo

NSERC

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